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Alternated hypercubic honeycomb

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An alternated square tiling or checkerboard pattern.
or
An expanded square tiling.

A partially filled alternated cubic honeycomb with tetrahedral and octahedral cells.
or
A subsymmetry colored alternated cubic honeycomb.

In geometry, the alternated hypercube honeycomb (or demicubic honeycomb) is a dimensional infinite series of honeycombs, based on the hypercube honeycomb with an alternation operation. It is given a Schläfli symbol h{4,3...3,4} representing the regular form with half the vertices removed and containing the symmetry of Coxeter group B ~ n − 1 {\displaystyle {\tilde {B}}_{n-1}} for n ≥ 4. A lower symmetry form D ~ n − 1 {\displaystyle {\tilde {D}}_{n-1}} can be created by removing another mirror on an order-4 peak.

The alternated hypercube facets become demihypercubes, and the deleted vertices create new orthoplex facets. The vertex figure for honeycombs of this family are rectified orthoplexes.

These are also named as hδn for an (n-1)-dimensional honeycomb.

hδn Name Schläfli
symbol 		Symmetry family
B ~ n − 1 {\displaystyle {\tilde {B}}_{n-1}}
D ~ n − 1 {\displaystyle {\tilde {D}}_{n-1}}
Coxeter-Dynkin diagrams by family
hδ2 Apeirogon {∞}
hδ3 Alternated square tiling
(Same as {4,4}) 		h{4,4}=t1{4,4}
t0,2{4,4}

hδ4 Alternated cubic honeycomb h{4,3,4}
{3,4}

hδ5 16-cell tetracomb
(Same as {3,3,4,3}) 		h{4,3,4}
{3,3,4}
{3}

hδ6 5-demicube honeycomb h{4,3,4}
{3,3,4}
{3,3,3}

hδ7 6-demicube honeycomb h{4,3,4}
{3,3,4}
{3,3,3}

hδ8 7-demicube honeycomb h{4,3,4}
{3,3,4}
{3,3,3}

hδ9 8-demicube honeycomb h{4,3,4}
{3,3,4}
{3,3,3}

 
hδn n-demicubic honeycomb h{4,3,4}
{3,3,4}
{3,3,3} 		...

References

  1. Regular and semi-regular polytopes III, p.318-319
  • Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
    1. pp. 122–123, 1973. (The lattice of hypercubes γn form the cubic honeycombs, δn+1)
    2. pp. 154–156: Partial truncation or alternation, represented by h prefix: h{4,4}={4,4}; h{4,3,4}={3,4}, h{4,3,3,4}={3,3,4,3}
    3. p. 296, Table II: Regular honeycombs, δn+1
  • Kaleidoscopes: Selected Writings of H. S. M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III,
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family A ~ n − 1 {\displaystyle {\tilde {A}}_{n-1}} C ~ n − 1 {\displaystyle {\tilde {C}}_{n-1}} B ~ n − 1 {\displaystyle {\tilde {B}}_{n-1}} D ~ n − 1 {\displaystyle {\tilde {D}}_{n-1}} G ~ 2 {\displaystyle {\tilde {G}}_{2}} / F ~ 4 {\displaystyle {\tilde {F}}_{4}} / E ~ n − 1 {\displaystyle {\tilde {E}}_{n-1}}
E Uniform tiling 0 δ3 hδ3 qδ3 Hexagonal
E Uniform convex honeycomb 0 δ4 hδ4 qδ4
E Uniform 4-honeycomb 0 δ5 hδ5 qδ5 24-cell honeycomb
E Uniform 5-honeycomb 0 δ6 hδ6 qδ6
E Uniform 6-honeycomb 0 δ7 hδ7 qδ7 222
E Uniform 7-honeycomb 0 δ8 hδ8 qδ8 133 • 331
E Uniform 8-honeycomb 0 δ9 hδ9 qδ9 152 • 251 • 521
E Uniform 9-honeycomb 0 δ10 hδ10 qδ10
E Uniform 10-honeycomb 0 δ11 hδ11 qδ11
E Uniform (n-1)-honeycomb 0 δn hδn qδn 1k2 • 2k1 • k21
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